for all \(h\)  in the Sobolev space \(H^1(p,w)\)  which we will define in Section REF . We abbreviate this by
\(\mathrm {PI}(p,w)\) . The smallest constant for which (REF ) holds
is called the Poincaré constant of \(p\)  with weight \(w\) , and denoted by
\(C(p,w)\) . When \(C(p,w)< \infty \) , we say that a test function \(h\)  saturates
\(\mathrm {PI}(p,w)\)  if (REF ) is an equality. Estimation of \(C(p,w)\)  and of the corresponding saturating functions is of theoretical importance because of its connections with concentration of measure (see [1]}, [2]}), isoperimetric inequalities (see [3]}, [4]}) and quantitative central limit theorems (see [5]}). This problem is also of practical importance, with uses ranging from the study of convergence properties of MCMC algorithms (see [6]}, [7]}, [8]}) to sensitivity analysis [9]}.
